Hereditary First-Order Logic

• Hereditary First-Order Logic is defined as the study of finite structures that hereditarily satisfy first-order sentences, focusing on induced substructures and logical expressibility.
• It classifies model checking tasks into polynomial-time and coNP-complete cases based on quantifier prefix patterns, highlighting clear boundaries of tractability.
• Its connections to logic programming and constraint satisfaction demonstrate practical implications in Prolog semantics and CSP optimization.

Hereditary First-Order Logic (HerFO) is the paper of properties of finite structures that are stable under taking induced substructures and that are definable by first-order sentences. The associated computational task, hereditary model checking (Her-MC), evaluates whether a finite structure’s every induced substructure satisfies a given first-order sentence $\phi$. This notion underpins a sharp complexity classification tied to the quantifier prefix of $\phi$, yielding a dichotomy between polynomial-time solvable and coNP-complete cases, subject to strong model-theoretic and complexity-theoretic constraints (Bodirsky et al., 16 Nov 2024). Additionally, the logic of hereditary Harrop formulas (fohh) extends this perspective to settings foundational in logic programming and computational semantics, as in Prolog (Kwon, 2015).

1. Formal Definition and Foundations

Let $\tau$ be a finite relational signature and $\phi$ a sentence of first-order logic (FO) over $\tau$. For a finite $\tau$-structure $A$, write $B \subseteq A$ for any induced substructure of $A$ (that is, $B$ retains the same universe elements and all $\tau$-facts inherited from $A$). A structure $A$ hereditarily satisfies $\phi$ if every induced substructure $B \subseteq A$ satisfies $\phi$.

$$\mathrm{Her}(\phi) := \{\, \text{finite } \tau\text{-structures } A : \forall B \subseteq A,\, B \models \phi \,\}$$

The hereditary model checking problem (Her-MC) for a fixed sentence $\phi$ is:

• Input: Finite structure $A$
• Question: Does $A \in \mathrm{Her}(\phi)$?

Since model checking $A \models \phi$ for fixed $\phi$ is in polynomial time, the hereditary problem lies in coNP: non-membership is witnessed by a counterexample substructure (Bodirsky et al., 16 Nov 2024).

2. Quantifier Prefix Classification and Tractability

Hereditary model checking exhibits a quantifier-prefix-based complexity dichotomy. For any quantifier prefix $Q \in \{\forall, \exists\}^*$ and $\phi$ in prenex normal form $Qx_1\ldots x_n.\psi$:

• If $Q$ is of the form $\forall^*\exists^*$ (universal block followed by existential block) or $\forall^*\exists\forall^*$, then $\mathrm{Her}(\phi)$ is decidable in polynomial time (even in $\mathrm{TC}^0$) and is FO-definable; hence, $$\mathrm{Her}(\phi) \in \mathrm{P} \cap \mathrm{SNP}$$.
• Otherwise—precisely, if $Q$ contains $\exists \exists \forall$ or $\exists \forall \exists$ as a subword—there exists some $\phi$ with this prefix such that $\mathrm{Her}(\phi)$ is coNP-complete (Bodirsky et al., 16 Nov 2024).

Tractable cases correspond exactly to the absence of the aforementioned existential-universal sandwich patterns, reflecting the preservation of expressibility and decidability.

Sketches of Tractability

• For $Q = \forall^* \exists^*$, e.g. $$\phi = \forall x_1\ldots x_k\, \exists y_1\ldots y_\ell.\psi(\bar{x},\bar{y})$$, a “small-model property” applies: $A\in\mathrm{Her}(\phi)$ iff every induced $k$-element substructure satisfies $\phi$, reducing to universal first-order checking and thus PTIME.
• For $Q = \forall^*\exists\forall^*$, the logic may be rewritten via greedy witness assignment or as monadic second-order logic. In both cases, $\mathrm{Her}(\phi)$ is in PTIME (Bodirsky et al., 16 Nov 2024).

3. Hardness of Prefixes and Typical coNP-complete Cases

If the quantifier prefix $Q$ contains a “forbidden” subword ($\exists\exists\forall$ or $\exists\forall\exists$), then $\mathrm{Her}(\phi)$ can realize coNP-completeness. For example, with $\tau = \{E,R\}$ (binary/ternary relations), formula

$$\eta := \exists x\,\exists y\,\forall a\; \left[\neg E(x,a) \lor R(x,y,a)\right]$$

enables reduction from the complement of 3-SAT to $\mathrm{Her}(\eta)$. Intuitively, failure to hereditarily satisfy $\eta$ can be witnessed by an induced substructure coding a certificate of unsatisfiability via cycles and triangles in the structure, mirroring coNP-hardness (Bodirsky et al., 16 Nov 2024). Similar constructions establish the hardness for any prefix featuring $\exists\exists\forall$ or $\exists\forall\exists$.

4. Undecidability of PTIME Membership for Arbitrary Formulas

Although the prefix classification provides a syntactic criterion for tractability, determining whether $\mathrm{Her}(\phi) \in \mathrm{P}$ is undecidable for arbitrary sentences $\phi$ unless $\mathrm{P} = \mathrm{NP}$. This follows from the ability to encode arbitrary existential second-order (ESO) problems in first-order logic, as well as the undecidability of FO prefix satisfiability beyond classical fragments (Bernays–Schönfinkel $\exists^*\forall^*$, Ackermann $\exists^*\forall\exists^*$). Thus, deciding hereditary tractability for arbitrary $\phi$ exceeds the known boundaries of algorithmic logic, with a direct equivalence to major open questions in complexity theory (Bodirsky et al., 16 Nov 2024).

5. Relationship to Constraint Satisfaction and Logic Programming

The class HerFO (hereditarily definable by FO) is situated between $\mathrm{P}$ and coNP, containing many natural PTIME CSPs (such as forests, 2-SAT, poset embeddability), but not capturing all of coNP unless $\mathrm{E} = \mathrm{NE}$. This provides a uniform framework encompassing:

• All Datalog-definable and bounded-treewidth CSPs (those corresponding to the $\forall^*\exists^*$ prefix)
• Guarded $\mu$-calculus style CSPs (those with $\forall^*\exists\forall^*$ prefixes)

Implications extend to logic programming. For example, first-order hereditary Harrop formulas (fohh) provide a type-theoretic generalization relevant to operational semantics in Prolog, with uniform provability (as in Miller et al.) and an alternative game-semantics execution model (Kwon, 2015). In the latter, universal quantification in goals corresponds to user-driven input (logically sanctioned “read”), yielding an interactive two-player game and connecting with computability logic frameworks.

6. Open Problems and Further Directions

Hereditary first-order logic raises several open research questions:

• Precise characterization of which finite-domain CSPs fall within HerFO.
• The status of CSPs for reducts of finitely bounded $\omega$-categorical structures with respect to HerFO and extensional SNP.
• Potential for a full constraint dichotomy within HerFO (beyond the current prefix dichotomy).
• Approximate and promise variants of hereditary model checking.

A plausible implication is that the connection between logical expressibility, substructure preservation, and complexity delineates a rich landscape in finite model theory, merging algebraic and algorithmic perspectives. Continued exploration is likely to yield deeper decompositions of the hereditary model-theoretic complexity classes and their interface with practical constraint problems (Bodirsky et al., 16 Nov 2024).